How Do You Solve a System of Linear ODEs with Equal Second Derivatives?

[s_j_sawyer]

Homework Statement

Solve this system of linear ODEs:

1) x''(t) = x + y
2) y''(t) = x + y

Just fyi, this is part of a much larger problem but I need to solve this system!

Homework Equations

See above.

The Attempt at a Solution

Okay so I think the most logical way to solve this would be to set x''(t) = y''(t).

Then
x''(t) = y''(t)
x' = y' + c1
x = y + c1t + c2

which implies

3) x'' = 2x - c1t - c2
4) y'' = 2y + c1t + c2

But I am not sure what to do from here. Apparently there should be 4 arbitrary constants in the final answer. But wouldn't solving x'' for x give you 2 and then solving y'' for y give you two more?

Thanks for any help.

 

[cronxeh]

Did you find the eigenvalues and eigenvectors? What is the general solution to this linear system of ODEs? Yes you will end up with 4 constants but that's the minor step

 

[Pere Callahan]

You could also write it as a vector equation
$$\vec x''(t) = \left(\begin{array}{cc} 1&1\\1&1\end{array}\right)\vec x(t)$$
and go on from there.